File size: 3,101 Bytes
f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | import OAI.Probability.FirstPassage.IntervalTravel
import OAI.Probability.FirstPassage.LocalizedEvents
namespace OAI
namespace PlanarFPP
open Set MeasureTheory Filter
open scoped Topology
def allPairsRegionEvent (V : Finset Vertex) (I J : Set Vertex)
(bound : Vertex → Vertex → ℝ) : Set (Edge → ℝ) :=
{w | ∀ p ∈ I, ∀ q ∈ J, w ∈ boundedLocalizedEvent V
(fun a ds => a=p ∧ a+displacement ds=q) (fun _ _ => bound p q)}
theorem mem_allPairsRegionEvent (V : Finset Vertex) (I J : Set Vertex)
(bound : Vertex → Vertex → ℝ) (w : Edge → ℝ) :
w ∈ allPairsRegionEvent V I J bound ↔
∀ p ∈ I, ∀ q ∈ J, ∃ ds, p+displacement ds=q ∧
PathIn p ds (V : Set Vertex) ∧ pathCost w p ds ≤ bound p q := by
simp [allPairsRegionEvent,boundedLocalizedEvent]
theorem measurableSet_allPairsRegionEvent (V : Finset Vertex) (I J : Set Vertex)
(bound : Vertex → Vertex → ℝ) : MeasurableSet (allPairsRegionEvent V I J bound) := by
simp only [allPairsRegionEvent,ofPred_forall]
exact MeasurableSet.iInter (fun p => MeasurableSet.iInter (fun _ =>
MeasurableSet.iInter (fun q => MeasurableSet.iInter (fun _ =>
measurableSet_boundedLocalizedEvent V _ _))))
theorem allPairsRegionEvent_depends (V : Finset Vertex) (I J : Set Vertex)
(bound : Vertex → Vertex → ℝ) :
DependsOnEdges (allPairsRegionEvent V I J bound) (internalEdges V) := by
intro w w' hh
simp only [allPairsRegionEvent,mem_ofPred_eq]
apply forall_congr'
intro p
apply forall_congr'
intro hp
apply forall_congr'
intro q
apply forall_congr'
intro hq
exact boundedLocalizedEvent_depends V _ _ w w' hh
variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ}
theorem exponential_allPairsRegionEvent (h : ExponentialEnvironment P τ)
(μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ)
(D η δ : ℝ) (hD : 0 < D) (hη : 0 < η) (hδ : 0 < δ) :
∀ ε > 0, ∀ᶠ s : ℝ in atTop, ∀ (V : Finset Vertex) (I J : Set Vertex)
(bound : Vertex → Vertex → ℝ),
(∀ p ∈ I ∪ J, ∀ q ∈ I ∪ J, ‖latticeEmbed (q-p)‖ ≤ D*s) →
(∀ p ∈ I, ∀ q ∈ J, μ (latticeEmbed (q-p))+η*s ≤ bound p q) →
(∀ p ∈ I, ∀ q ∈ J, SegmentTube p q (δ*s) ⊆ (V : Set Vertex)) →
P.real {ω | (fun e => τ e ω) ∉ allPairsRegionEvent V I J bound} < ε := by
let _ := h.probability
intro ε hε
filter_upwards [exponential_set_travel h μ hμ ht D η δ hD hη hδ ε hε] with s hs
intro V I J bound hdiam hcost hregion
refine (measureReal_mono (show {ω | (fun e => τ e ω) ∉ allPairsRegionEvent V I J bound} ⊆
{ω | ¬∀ p ∈ I, ∀ q ∈ J, LocalTravel μ η δ s p q (fun e => τ e ω)} from ?_)).trans_lt
(hs I J hdiam)
intro ω hbad htravel
apply hbad
rw [mem_allPairsRegionEvent]
intro p hp q hq
obtain ⟨ds,hd,hc,hin⟩ := htravel p hp q hq
refine ⟨ds,hd,?_,hc.trans (hcost p hp q hq)⟩
intro i hi
exact hregion p hp q hq (hin i hi)
end PlanarFPP
end OAI
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